{tf S}$^{bold 1}${td X}{tf S}$^{bold 1}$-INDEX THEORY ONPRODUCT SPACE

A new index is constructed by use of the canonical representationof $S^1times S^1$ group over a product space. This indexsatisfies the general properties of the usual index but does notsatifsy the dimension property. As an application, two abstractcritical point theorems are given.     

Tail indices for $$\mathbf{A}\mathbf{X}+\mathbf{B}$$ A X + B Recursion with Triangular Matrices

Journal of Theoretical Probability - We study multivariate stochastic recurrence equations (SREs) with triangular matrices. If coefficient matrices of SREs have strictly positive entries, the...     

Positive definite solution of the matrix equation boldsymbol {X=Q+A^{H}(Iotimes X-C)^{delta}A}

Consider the nonlinear matrix equation X?=?Q?+?A ^H (I???X???C) ^?? A ( ???=???1 or 0?<?|??|?<?1), where Q is an n×n positive definite matrix, C is an mn ×mn positive semidefinite matrix, I is an m×m identity matrix, and A is an arbitrary mn×n matrix. This equation is connected with a certain interpolation problem when ???=???1. Using the properties of the Kronecker product and the theory for the monotonic operator defined in a normal cone, we prove the existence and uniqueness of the positive definite solution which is contained in the set {X|I???X?>?C} under the condition that I???Q?>?C. The iterative methods to compute the unique solution is proposed. Numerical examples show that the methods are feasible and effective.     

On the Matrix Equation X^{ - 1} X* = A

Solvability conditions are examined for the matrix equation , which cannot be found in the well-known reference books on matrix theory. Methods for constructing solutions to this equation are indicated.     

On a Solution of the Quaternion Matrix Equation {A \tilde{X} -X B = C} and Its Applications

This paper studies the problem of solution of the quaternion matrix equation ${A \tilde{X} -X B = C}$ by means of real representation of a quaternion matrix, derives a similarity version of Roth’s theorem for the existence of solution to the quaternion matrix equation, and obtains closed-form solutions of the quaternion matrix equation in explicit forms. As a special case, this paper also gives some applications of the solutions of complex matrix equation ${A \bar{X} -X B = C}$ and of the consimilarity of complex matrices.     

Commuting involution graphs for \tilde{A}_{n}

In this article we consider the commuting graphs of involution conjugacy classes in the affine Weyl group We show that where the graph is connected the diameter is at most 6. Received: 24 February 2005     

Isometric embeddings of \mathbb{Z}_{p^k} in the Hamming space \mathbb{F}_{p}^{N} and \mathbb{Z}_{p^k}-linear codes

Isometric embeddings of $\mathbb{Z}_{p^n+1}$ into the Hamming space ( $\mathbb{F}_{p}^{p^n},w$ ) have played a fundamental role in recent constructions of non-linear codes. The codes thus obtained are very good codes, but their rate is limited by the rate of the first-order generalized Reed–Muller code—hence, when n is not very small, these embeddings lead to the construction of low-rate codes. A natural question is whether there are embeddings with higher rates than the known ones. In this paper, we provide a partial answer to this question by establishing a lower bound on the order of a symmetry of ( $\mathbb{F}_{p}^{N},w$ ).     

Geometric Optimization Problems Likely Not Contained in $$\mathbb{A}\mathbb{P}\mathbb{X}$$

   Abstract. Maximizing geometric functionals such as the classical l _p -norms over polytopes plays an important role in many applications, hence it is desirable to know how efficiently the solutions can be computed or at least approximated. While the maximum of the l _∞ -norm over polytopes can be computed in polynomial time, for 2≤ p < ∞ the l _p -norm-maxima cannot be computed in polynomial time within a factor of 1.090 , unless P=NP. This result holds even if the polytopes are centrally symmetric parallelotopes. Quadratic Programming is a problem closely related to norm-maximization, in that in addition to a polytope P ⊂ R ⁿ , numbers c _ij , 1≤ i≤ j≤ n , are given and the goal is to maximize Σ _{1≤ i≤ j≤ n} c _ij x _i x _j over P . It is known that Quadratic Programming does not admit polynomial-time approximation within a constant factor, unless P=NP. Using the observation that the latter result remains true even if the existence of an integral optimal point is guaranteed, in this paper it is proved that analogous inapproximability results hold for computing the l _p -norm-maximum and various l _p -radii of centrally symmetric polytopes for 2≤ p < ∞.     

Factorizations of one-generated $$ \mathfrak{X} $$-local formations

Under study are the factorizations of one-generated \(\mathfrak{X}\)-local formations, where \(\mathfrak{X}\) is a class of simple groups. Some known results on one-generated local and Baer-local formations appear as particular cases.     

Iterative algorithm for solving a class of general Sylvester-conjugate matrix equation \sum_{i = 1}^{s} A_{i}V + \sum_{j = 1}^{t} B_{j}W = \sum_{l = 1}^{m} E_{l}\overline{V}F_{l} + C

This paper is concerned with iterative solution to general Sylvester-conjugate matrix equation of the form $\sum_{i = 1}^{s} A_{i}V + \sum_{j = 1}^{t} B_{j}W = \sum_{l = 1}^{m} E_{l}\overline{V}F_{l} + C$ . An iterative algorithm is established to solve this matrix equation. When this matrix equation is consistent, for any initial matrices, the solutions can be obtained within finite iterative steps in the absence of round off errors. Some lemmas and theorems are stated and proved where the iterative solutions are obtained. Finally, a numerical example is given to verify the effectiveness of the proposed algorithm.     

$$ H^{p}_{w} - L^{p}_{w} $$ Boundedness of Marcinkiewicz Integral

The Marcinkiewicz integral is essentially a Littlewood-Paley g-function, which plays a very important role in harmonic analysis. In this paper we give weaker smoothness conditions assumed on Ω to imply the boundedness of the Marcinkiewicz integral operator μ_Ω, where w belongs to the Muckenhoupt weight class.